Solve For \[$ X \$\].$\[ \left(\frac{1}{4}\right)^{2-\sqrt{5x+1}} = 4 \times (2)^{\sqrt{5x+1}} \\]

Feb 28, 2025 by ADMIN 99 views

Solving Exponential Equations with Variable Bases and Exponents

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Introduction

Exponential equations with variable bases and exponents can be challenging to solve. In this article, we will focus on solving an exponential equation that involves a fraction as the base and a variable exponent. The equation is given as:

$\left(\frac{1}{4}\right)^{2-\sqrt{5x+1}} = 4 \times (2)^{\sqrt{5x+1}}$

Our goal is to isolate the variable $x$ and find its value.

Step 1: Simplify the Right-Hand Side of the Equation

To simplify the right-hand side of the equation, we can rewrite $4$ as $2^2$ . This gives us:

$\left(\frac{1}{4}\right)^{2-\sqrt{5x+1}} = 2^2 \times (2)^{\sqrt{5x+1}}$

Using the properties of exponents, we can combine the two terms on the right-hand side:

$\left(\frac{1}{4}\right)^{2-\sqrt{5x+1}} = 2^{2+\sqrt{5x+1}}$

Step 2: Rewrite the Left-Hand Side of the Equation

We can rewrite the left-hand side of the equation by expressing $\frac{1}{4}$ as $2^{-2}$ . This gives us:

$\left(2^{-2}\right)^{2-\sqrt{5x+1}} = 2^{2+\sqrt{5x+1}}$

Using the property of exponents that states $(a^m)^n = a^{mn}$ , we can simplify the left-hand side:

$2^{-2(2-\sqrt{5x+1})} = 2^{2+\sqrt{5x+1}}$

Step 3: Equate the Exponents

Since the bases are the same, we can equate the exponents:

$-2(2-\sqrt{5x+1}) = 2+\sqrt{5x+1}$

Step 4: Simplify the Equation

We can simplify the equation by distributing the $-2$ :

$-4 + 2\sqrt{5x+1} = 2+\sqrt{5x+1}$

Step 5: Isolate the Variable

We can isolate the variable $x$ by subtracting $\sqrt{5x+1}$ from both sides:

$-4 = 2 + 2\sqrt{5x+1} - \sqrt{5x+1}$

Simplifying further, we get:

$-6 = \sqrt{5x+1}$

Step 6: Square Both Sides

To eliminate the square root, we can square both sides of the equation:

$(-6)^2 = (\sqrt{5x+1})^2$

This gives us:

$36 = 5x+1$

Step 7: Solve for $x$

We can solve for $x$ by subtracting $1$ from both sides:

$35 = 5x$

Dividing both sides by $5$ , we get:

$x = \frac{35}{5}$

Simplifying further, we get:

$x = 7$

The final answer is $\boxed{7}$ .

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Q: What is an exponential equation with a variable base and exponent?

A: An exponential equation with a variable base and exponent is an equation that involves a base that is a variable and an exponent that is also a variable. In the equation $\left(\frac{1}{4}\right)^{2-\sqrt{5x+1}} = 4 \times (2)^{\sqrt{5x+1}}$ , the base is $\frac{1}{4}$ and the exponent is $2-\sqrt{5x+1}$ .

Q: How do I simplify an exponential equation with a variable base and exponent?

A: To simplify an exponential equation with a variable base and exponent, you can start by rewriting the equation in a more manageable form. This may involve expressing the base as a power of a simpler base, or using the properties of exponents to combine terms.

Q: What is the first step in solving an exponential equation with a variable base and exponent?

A: The first step in solving an exponential equation with a variable base and exponent is to simplify the equation by rewriting the base and exponent in a more manageable form. This may involve using the properties of exponents to combine terms, or expressing the base as a power of a simpler base.

Q: How do I equate the exponents in an exponential equation with a variable base and exponent?

A: To equate the exponents in an exponential equation with a variable base and exponent, you can set the exponents equal to each other and solve for the variable. This may involve using algebraic manipulations to isolate the variable.

Q: What is the final step in solving an exponential equation with a variable base and exponent?

A: The final step in solving an exponential equation with a variable base and exponent is to solve for the variable by isolating it on one side of the equation. This may involve using algebraic manipulations to isolate the variable.

Q: What are some common mistakes to avoid when solving exponential equations with variable bases and exponents?

A: Some common mistakes to avoid when solving exponential equations with variable bases and exponents include:

Failing to simplify the equation before solving for the variable
Failing to equate the exponents correctly
Failing to isolate the variable correctly
Making algebraic errors when solving for the variable

Q: How can I practice solving exponential equations with variable bases and exponents?

A: You can practice solving exponential equations with variable bases and exponents by working through example problems and exercises. You can also try solving more complex equations to challenge yourself and build your skills.

Q: What are some real-world applications of exponential equations with variable bases and exponents?

A: Exponential equations with variable bases and exponents have many real-world applications, including:

Modeling population growth and decay
Modeling financial growth and decay
Modeling chemical reactions and decay
Modeling physical systems and decay

Q: How can I use technology to solve exponential equations with variable bases and exponents?

A: You can use technology, such as graphing calculators or computer software, to solve exponential equations with variable bases and exponents. This can be especially helpful when working with complex equations or when you need to visualize the solution.

Q: What are some tips for solving exponential equations with variable bases and exponents?

A: Some tips for solving exponential equations with variable bases and exponents include:

Simplifying the equation before solving for the variable
Equating the exponents correctly
Isolating the variable correctly
Checking your work to ensure that the solution is correct

By following these tips and practicing regularly, you can become proficient in solving exponential equations with variable bases and exponents.